The Lennard-Jones potential and the Pauli exclusion principle The force between two atoms takes the generic shape of a Lennard-Jones potential. It has an attractive part caused by dipole-dipole attraction and a short-range repulsive part which is said to be caused by Pauli's exclusion principle. 
Can some one explain rigorously how the repulsive part of the potential arises from Pauli exclusion principle? 
 A: The Lennard-Jones potential is $$U(r) = A r^{-12} - B r^{-6}.$$The function of the first term is to make the energy go to $+\infty$ as $r\to 0$ as a way of  modeling Pauli exclusion with a potential barrier that stops two particles from being in the same location. In fact this term is purely heuristic, and any power $n>6$ would work, but higher powers have the advantage that we expect the actual long-range Pauli effect to decay like $e^{-kr}$ as $r$ gets larger, since when we solve for the wavefunction of the Hydrogen atom we find that the wavefunction of the electron cloud decays radially like $e^{-r/2r_\text{Bohr}}$ and the overlap of the two electron clouds has to be the source of the Pauli exclusion principle; this must decay faster than $r^{-n}$ for any $n$. So we prefer higher powers as they decay to 0 faster and therefore introduce less long-range weirdness, while still allowing for a repulsive effect that keeps the atoms from falling into each other.
The reason $r^{-12}$ is used in practice is that it is a very high $n$ which can be gotten from the $r^{-6}$ that one already has to calculate (which can be explained as the London forces) by one single multiplication instruction; $r^{-6}\cdot r^{-6} = r^{-12}.$ 
The 1924 paper by John Lennard-Jones actually considered a wide range of $n$ and $m$ for potentials of the form $U(r) = A r^{-n} - B r^{-m},$ trying to match his results against the measured viscosity of liquid argon. He found that good fitting of this potential required $m=6$ (the paper actually discusses forces so it says $m=5$ but this integrates to an $m=6$ potential), but that a lot of different $n$ seemed to be valid choices, and any $n > 10$ worked quite well, with probably his best results for this particular experiment of $n = 15 + 1/3.$ So going up another power to $r^{-18}$ or $r^{-24}$ does not seem to "buy us much" over $r^{-12}$ and we just use that in practice.
A: Really nice question, wich is rarely arise.
In fact the answer is I guess :
If you determine the true wave function of your molecule by solving the true hamiltonian of the two well potential + electronic potential, you get the true result which is very close of the results you get using a Lennard-Jones potential.
The Pauli exclusion principle is include in the quantum mechanic !
Where ? 
It is not a fundamental principe of quantum mechanic, it is derived from the commutation of the hamiltonien with the exchange operator $P$ . If the hamiltonian $H$ commute with $P$ then you have two familly of solution : symmetric and anti-symmetric. When solving the hamilotnian for electron you explicitly keep only anti-symetric part. And you can easly demonstrate that anti-symmetric functions follow Pauli exclusion principle !
So it is by explicitly keeping only anti-symmetric solutions that you insure the Pauli Exclusion Principle.
But you can't show ( i'm not sure) that the Lennard-Jones potential describe well this phenomenone. It is totally empiric way but which work
